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# Mathematical Proof that Regular Languages are Closed under Reversal

• Last Updated : 02 Feb, 2023

A language is regular if it can be expressed in terms of regular expressions. The article focuses on discussing the mathematical proof of the fact that Regular Languages are closed under reversal.

Regular Languages generally have three basic definitions:

1. Regular Languages are those Languages that have a Regular Expression to represent them.
2. Regular Languages are those Languages that have a regular grammar defined on them.
3. Regular languages are those languages for which a finite automaton can be created.

We know that these regular languages are closed under union, reversal, etc.

Proof:

Let L be a  regular language, and M be an NFA that accepts it.

M = (Q, Î£, Î´, q0, {qf})

Here,

Q = Set of states in the finite automata
Î£ = alphabet
Î´ = Transition
q0 = Initial State
F = {qf} = Set of final States

We construct MR

MR = (Q, Î£, Î´R, qf , {q0})

Here,
Î´R is Î´ with the direction of all the arcs reversed.

There is a path from q0 to qf in M if and only if there is a path from qf to q0 in MR.

This implies that:

L(MR) = LR

Thus, it is proved that L is closed under reversal.

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